Overcoming Globally Inoptimal Strategies through Repeated Games of Trust
https://ncase.me/trust/
This is an interactive article written by computer scientist Nicki Case. Despite its game-like appearance (pun intended) the article touches on some very advanced topics in game theory. Case begins by introducing a coin game very much in the spirit of Prisoner’s Dilemma. There are 2 players. Each player has the option to put or not put one of his own coins in a machine. When a player puts a coin in the machine gives his opponent gets 3 coins (and vice versa). Thus:
- When both players put coins in the payoffs are (+2.+2)
- When one player puts a coin in while the other doesn’t the payoffs are (+3.-1)
- When neither player puts a coin in, the payoffs are (0,0)
One key difference in the coin game is that player plays multiple rounds with each other, before moving on to another player.
Case makes the obvious point that the dominant strategy is to never put a coin in the machine. However, he also points out that the best strategy is not “Always Cheat”, but rather “Copycat” (start off by attempting to cooperate, then respond to an opponent’s cheat with a cheat and responding to an opponent’s cooperation with cooperation). Copycats can achieve the globally optimum when playing with perpetual cooperators, while still avoid being swindled by perpetual cheaters.
Case then goes on to explain the disparity between the “Always Cheat” strategy that makes sense and the “Copycat” strategy that works. He emphasizes the “repeated game of trust” and later factors in mistakes to the game (ie, a player wants to cooperate but pushes the wrong button and cheats).
Analysis:
While Case doesn’t use the exact terms in his article, several topics covered in class appear throughout his article. The “Always Cheat” strategy results in a Nash equilibrium. However, as we noted in class the Nash equilibrium is not always the global optimum. In this case, the global optimum would be for both players to put in a coin and receive a payoff of (+2,+2). The Nash Equilibrium of both players cheating is a local optimum, of sorts.
However, Case’s coin game differs from the Prisoner’s Dilemma in that the game is played for multiple rounds between players. This introduces some new dynamics to the game that we haven’t covered in class (yet?). Case points out 3 factors that must work together to transition the population’s to the “Copycat” strategy:
- Repeated games of trust. Trust evolves over time. In a 1-off game like Prisoner’s Dilemma, trust and cooperation are irrelevant.
- Possible win-wins. In the game, mutual cooperation is the global optimum. In order for any cooperation to occur, there has to be some incentive (ie the global optimum)
- Low mistakes. Mistakes (mentioned in summary) can cause problems in building trust.
It is hard to get past the intuition that in repeated games, “Always Cheat” is still the best strategy. We’ve begun to explore the idea of repeated games with our lectures on mixed strategies; however, much of what is discussed in this article is slightly beyond the current scope of the course. Nevertheless, it will be interesting to see how this idea of population/aggregate behavior comes up in later units.